Polynomial Linear System Solving with Errors by Simultaneous - PowerPoint PPT Presentation

Polynomial Linear System Solving with Errors by Simultaneous Polynomial Reconstruction of Interleaved Reed-Solomon Codes. Ilaria Zappatore a joint work with Eleonora Guerrini LIRMM, Montpellier Journées Nationales de Calcul Formel 2019 CIRM, Luminy.

Table of contents 1. Polynomial linear system solving The evaluation/interpolation method Polynomial linear system solving with errors Generalization of BW decoding for Reed Solomon codes 2. Generalization of the decoding of interleaved RS codes Interleaved Reed Solomon codes Decoding interleaved RS codes Our method 3. Experiments and conclusions 1

Polynomial linear system solving

Polynomial linear system where, 2 a full rank consistent polynomial linear system Fixed a fjnite fjeld F q , m ≥ n ≥ 1, we consider the problem of solving A ( x ) y ( x ) = b ( x )       a 1 , 1 ( x ) a 1 , 2 ( x ) . . . a 1 , n ( x ) y 1 ( x ) b 1 ( x )       a 2 , 1 ( x ) a 2 , 2 ( x ) . . . a 2 , n ( x ) y 2 ( x ) b 2 ( x )       =  . . . .   .   .  . . . . . .       . . . . . .       a m , 1 ( x ) a m , 2 ( x ) . . . a m , n ( x ) y n ( x ) b m ( x ) • A ( x ) is a full rank m × n matrix whose entries are polynomials in F q [ x ] , • b ( x ) is an m-th vector of polynomials in F q [ x ] .

Polynomial linear system (1) Our aim is to fjnd the polynomials f and g such that 3 There is a unique rational solution a full rank consistent polynomial linear system Fixed a fjnite fjeld F q , m ≥ n ≥ 1, we consider the problem of solving A ( x ) y ( x ) = b ( x )   f 1 ( x ) g ( x )   f 2 ( x ) y ( x ) := f ( x )   g ( x )   g ( x ) = .   .  .    f n ( x ) g ( x ) where g ( x ) is the monic least common denominator and GCD ( f , g ) = GCD ( GCD i ( f i ) , g ) = 1 . A ( x ) f ( x ) = g ( x ) b ( x ) .

Polynomial linear system There is a unique rational solution Our aim is to fjnd the polynomials f and g such that (1) 3 a full rank consistent polynomial linear system Fixed a fjnite fjeld F q , m ≥ n ≥ 1, we consider the problem of solving A ( x ) f ( x ) g ( x ) = b ( x )   f 1 ( x ) g ( x )   f 2 ( x ) y ( x ) := f ( x )   g ( x )   g ( x ) = .   .  .    f n ( x ) g ( x ) where g ( x ) is the monic least common denominator and GCD ( f , g ) = GCD ( GCD i ( f i ) , g ) = 1 . A ( x ) f ( x ) = g ( x ) b ( x ) .

l b Evaluation/Interpolation l • interpolating from the evaluated solution the parametric one. L 1 l 0 l l n 2 l 1 l A • solving the evaluating system we can uniquely reconstruct f and g by 4 Fix L ≥ df + dg + 1 distinct evaluation points { α 1 , α 2 , . . . , α L } , where • df ≥ max 1 ≤ i ≤ n deg( f i ) , • dg ≥ deg( g ) . • evaluating the polynomial matrix A ( x ) and b ( x ) at α l , 1 ≤ l ≤ L

Evaluation/Interpolation • solving the evaluating system • interpolating from the evaluated solution the parametric one. 4 we can uniquely reconstruct f and g by Fix L ≥ df + dg + 1 distinct evaluation points { α 1 , α 2 , . . . , α L } , where • df ≥ max 1 ≤ i ≤ n deg( f i ) , • dg ≥ deg( g ) . • evaluating the polynomial matrix A ( x ) and b ( x ) at α l , 1 ≤ l ≤ L     ϕ 1 ( α l )   ϕ 2 ( α l )       A ( α l ) − ψ ( α l ) b ( α l ) = 0     . .     .     ϕ n ( α l ) l ∈{ 1 ,..., L }

Evaluation/Interpolation • solving the evaluating system • interpolating from the evaluated solution the parametric one. 4 we can uniquely reconstruct f and g by Fix L ≥ df + dg + 1 distinct evaluation points { α 1 , α 2 , . . . , α L } , where • df ≥ max 1 ≤ i ≤ n deg( f i ) , • dg ≥ deg( g ) . • evaluating the polynomial matrix A ( x ) and b ( x ) at α l , 1 ≤ l ≤ L     ϕ 1 ( α l )   ϕ 2 ( α l )       A ( α l ) − ψ ( α l ) b ( α l ) = 0     . .     .     ϕ n ( α l ) l ∈{ 1 ,..., L }

Polynomial linear system solving with errors 1 We omit the rank drops study. Erroneous evaluation 5 =A( 흰 l ) A l ≠ A( 흰 l ) 흰 l b l =b( 흰 l ) ≠ b( 흰 l ) An evaluation point α l is erroneous if A l f ( α l ) � = g ( α l ) b l E := |{ l | A l f ( α l ) � = g ( α l ) b l }| . Since A l is full rank 1 for any l , A l f ( α l ) � = g ( α l ) b l = ⇒ A l � = A ( α l ) or/and b l � = b ( α l ) .

Polynomial linear system solving with errors How many evaluation points? [BK14] and [Kal+17] proved that with evaluation points , it is possible to uniquely reconstruct f and g . 6 L ≥ L BK := df + dg + 2 e + 1 • df ≥ max 1 ≤ i ≤ n deg( f i ) , • dg ≥ deg( g ) , • e ≥ | E | := |{ l | A l f ( α l ) � = g ( α l ) b l }|

A l f l e . dg e and deg df where deg b l l l l g l l Polynomial linear system solving with errors Main idea L , 1 • we put for any l e ; that is monic and has degree deg l x l E be the error locator polynomial , • let 7 • For any correct evaluations we have A l f ( α l ) = g ( α l ) b l

A l f Polynomial linear system solving with errors l e . dg e and deg df where deg b l l l l g l l Main idea L , 1 • we put for any l 7 • For any correct evaluations we have A l f ( α l ) = g ( α l ) b l • let Λ be the error locator polynomial , � Λ := ( x − α l ) , l ∈ E that is monic and has degree deg (Λ) ≤ e ;

Polynomial linear system solving with errors Main idea b l 7 • For any correct evaluations we have A l f ( α l ) = g ( α l ) b l • let Λ be the error locator polynomial , � Λ := ( x − α l ) , l ∈ E that is monic and has degree deg (Λ) ≤ e ; • we put for any l ∈ { 1 , . . . , L } , A l f ( α l )Λ( α l ) = g ( α l )Λ( α l ) � �� � � �� � ϕ ( α l ) ψ ( α l ) where deg ( ϕ ) ≤ df + e and deg ( ψ ) ≤ dg + e .

Polynomial linear system solving with errors Theorem [BK14] minimal degree of all such solutions. Then 8 Assume that • the number of erroneous evaluations is ≤ e , • the number of the correct evaluations for which A l is full rank is ≥ df + dg + e + 1 Let ( ϕ min , ψ min ) be a solution of  A 1 ϕ ( α 1 ) − ψ ( α 1 ) b 1 = 0     . . .    A L ϕ ( α L ) − ψ ( α L ) b L = 0  where ψ min is scaled to have leading coeffjcient 1 in x and it has ϕ min = Λ f , ψ min = Λ g

Reed Solomon Codes The Reed Solomon code is Maximum Distance Separable (MDS), i.e. 2 it matches the Singleton bound. Its error correction capability is 9 Let F q be a fjnite fjeld. Fixed: • df < L ≤ q , • L evaluation points , { α 1 , . . . , α L } , The Reed Solomon Code of length L and dimension df + 1 is the set RS q := { ( f ( α 1 ) , . . . , f ( α L )) | f ∈ F q [ x ] , deg ( f ) ≤ df } . ∑ 0 ≤ i ≤ df f i x i f 0 f 1 … f df f( 흰 1 ) f( 흰 2 ) f( 흰 3 ) f( 흰 4 ) … f( 흰 L ) e RS ≤ L − df − 1

Polynomial linear system solving with errors The [BK14] method is a generalization of the Berlekamp-Welch decoding for Reed Solomon codes. Recover the solution of the polynomial linear system Decoding of Reed Solomon code 10 If m = n = 1, A = I 1 , g constant polynomial 1, � b 1 b 2 b 3 b 4 … b L

Polynomial linear system solving with errors The [BK14] method is a generalization of the Berlekamp-Welch decoding for Reed Solomon codes. Recover the solution of the polynomial linear system Decoding of Reed Solomon code 10 If m = n = 1, A = I 1 , g constant polynomial 1, � f( 흰 1 ) f( 흰 3 ) f( 흰 L ) …

Generalization of the decoding of interleaved RS codes

Our approach Our purpose is to reconstruct the solution using a technique, inspired by the [BKY03] decoding of interleaved RS codes . 11 A(x) f( x)=g(x)b(x) =A( 흰 l ) A l ≠ A( 흰 l ) Boyer et al. (2014) 흰 l Kaltofen et al. b l =b( 흰 l ) (2017) ≠ b( 흰 l ) Berlekamp-Welch decoding Reed Solomon codes

Our approach Our purpose is to reconstruct the solution using a technique, inspired by the [BKY03] decoding of interleaved RS codes . 11 A(x) f( x)=g(x)b(x) =A( 흰 l ) A l ≠ A( 흰 l ) Boyer et al. (2014) 흰 l Kaltofen et al. This work b l =b( 흰 l ) (2017) ≠ b( 흰 l ) Berlekamp-Welch decoding Decoding of interleaved Reed Solomon codes Reed Solomon codes

Interleaved RS codes 12 n codewords of RS[L,df+1] q f 1 ( 𝝱 1 ) f 1 ( 𝝱 2 ) f 1 ( 𝝱 3 ) f 1 ( 𝝱 4 ) … f 1 ( 𝝱 L ) … f 2 ( 𝝱 1 ) f 2 ( 𝝱 2 ) f 2 ( 𝝱 3 ) f 2 ( 𝝱 4 ) f 2 ( 𝝱 L ) f 3 ( 𝝱 1 ) f 3 ( 𝝱 2 ) f 3 ( 𝝱 3 ) f 3 ( 𝝱 4 ) … f 3 ( 𝝱 L ) … f n ( 𝝱 1 ) f n ( 𝝱 2 ) f n ( 𝝱 3 ) f n ( 𝝱 4 ) f n ( 𝝱 L )

Interleaved RS codes 12 n codewords of RS[L,df+1] q f 1 ( 𝝱 1 ) f 1 ( 𝝱 2 ) f 1 ( 𝝱 3 ) f 1 ( 𝝱 4 ) … f 1 ( 𝝱 L ) … f 2 ( 𝝱 1 ) f 2 ( 𝝱 2 ) f 2 ( 𝝱 3 ) f 2 ( 𝝱 4 ) f 2 ( 𝝱 L ) f ( 𝝱 1 ) f 3 ( 𝝱 1 ) f 3 ( 𝝱 2 ) f 3 ( 𝝱 3 ) f 3 ( 𝝱 4 ) … f 3 ( 𝝱 L ) … f n ( 𝝱 1 ) f n ( 𝝱 2 ) f n ( 𝝱 3 ) f n ( 𝝱 4 ) f n ( 𝝱 L ) f ( 𝝱 1 ) ∈ ( 𝔾 q ) n